continuous random variables

A supermarket manager finds that 20% of his customers buy brand X cornflakes. The other 80% buy brand Z.

a) Ina ramdom sample of 10, what is the probability that at least 4 customers buy brand X? [ANS:0.121]

b) In a random sample of 100 customers, what is the probability that between 15 and 30 customers (inclusively) buy brand X? [ANS:0.9119]

c) What is the minimum number of boxes of brand X cornflakes that the manager should have in stock to ensure, with a probability of at least 0.95, that he can meet the brand X demands of a random sample of 225 customers? (Assume 1 box per customer) [ANS:55]

A supermarket manager finds that 20% of his customers buy brand X cornflakes. The other 80% buy brand Z.

c) What is the minimum number of boxes of brand X cornflakes that the manager should have in stock to ensure, with a probability of at least 0.95, that he can meet the brand X demands of a random sample of 225 customers? (Assume 1 box per customer) [ANS:55]

The critical z-score for a standard normal distribution, so that the probability
of not-exceeding it is is .

With a sample size of costomers the mean number who buy brand X is, and the standard devistion is .

So the number of customers corresponding to the critical z-score satisfies: